The P4 -structure of perfect graphs - Semantic Scholar

This paper appeared as a chapter in the book: Perfect Graphs, 93-112 , John Wiley & Sons, Inc. 2001

The P4-structure of perfect graphs Stefan Hougardy Humboldt-Universität zu Berlin Institut f¨ ur Informatik 10099 Berlin, GERMANY [email protected]

July 11th, 2000

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Introduction

One of the most important outstanding open problems in algorithmic graph theory is to determine the complexity of recognizing perfect graphs. Results of Lovász [33], Padberg [37] and Bland et al. [6] imply, as was first observed by Cameron [10] in 1982, that the problem of recognizing perfect graphs is in co-NP. So far it is not known whether this problem also belongs to NP, i.e., we do not know of any reasonable way to certify the perfection of an arbitrary graph. One weak form of such a certificate is obtained via the Perfect Graph Theorem: to prove the perfection of a graph it is enough to show that its complement is perfect. In attempting to generalize this kind of certificate, Chvátal [11] invented in 1984 the notion of P4 -structure. For a given graph G = (V, E), its P 4 -structure is defined as the 4-uniform hypergraph on V (G) whose edges are all the 4-element sets that induce a P4 (i.e., a path on four vertices) in G. Chvátal [11] conjectured that the perfection of a graph depends solely on its P 4 structure. He was led to this conjecture by observing that odd cycles and their complements have unique P4 -structure, i.e., any graph whose P 4 -structure is isomorphic (as a hypergraph) to the P4 -structure of an odd cycle is an odd cycle or its complement. Therefore the truth of the Strong Perfect Graph Conjecture would imply his conjecture. Moreover, as the P4 is a self-complementary graph, the P 4 -structure of a graph and its complement are isomorphic. This shows that Chvátal’s conjecture 1


implies the Perfect Graph Theorem. Chvátal therefore suggested his conjecture be called the Semi Strong Perfect Graph Conjecture. In 1987, Reed [40] proved the conjecture and so it is now known as the Semi Strong Perfect Graph Theorem. Meanwhile we have a fairly good knowledge of the P 4 -structure of perfect and minimally imperfect graphs. It turns out that P 4 -structure seems not to be powerful enough to get an NP-characterization for perfect graphs. Nevertheless the notion of P4 -structure has turned out to be a fruitful concept leading to many interesting structural and algorithmic results, not only in the area of perfect graphs. In Section 2 we introduce the notion of P 4 -structure and P4 -isomorphism and discuss the problem of recognizing the P 4 -structures of graphs which is the central problem in this area. Section 3 introduces the notions of modules and homogeneous sets and their counterparts in hypergraphs called h-sets. The powerful tools of modular and homogeneous decompositions allow the efficient solution of the recognition problem and some optimization problems for several classes of graphs. Moreover, they suggest a technique for solving the P 4 -structure recognition problem and indicate why the P4 -structures of split graphs are of special interest in this context. Section 4 discusses the Semi Strong Perfect Graph Theorem, its consequences for the problem of recognizing perfect graphs, and its connection to the problem of recognizing the P4 -structures of graphs. In Section 5 we study graphs that have unique respectively strongly unique P 4 -structure. Based on the results in this section a polynomial time algorithm to recognize P 4 -structures of graphs is outlined in Section 6. Also a survey on the recognition of the P 4 -structure of special classes of (perfect) graphs is given in this section. Section 7 surveys results on the P 4 -structure of minimally imperfect graphs. Section 8 contains some extensions of the notion of P4 -structure and Section 9 shows that several results proven for P 4 -structure also hold for P3 -structure.

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P4 -Structure: Basics, Isomorphisms and Recognition

The P4 -structure of a graph G = (V, E) is a 4-uniform hypergraph on V (G) whose hyperedges are all sets on four vertices that induce a P 4 (a path on four vertices) in G. Figure 1 shows an example of a graph and its P 4 -structure. Note that the P4 structure of a graph contains only the information which vertices in the graph induce a P4 , but no information how these four vertices are connected to induce the P 4 . If for a 4-uniform hypergraph H there exists a graph G such that H is the P 4 -structure of G, then we call G a realization of H. We say that a hypergraph H has a unique realization, if it has exactly one realization or if it has exactly two realizations which

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Figure 1: A graph and its P4 -structure. Two graphs are said to have the same P 4 -structure if their P4 -structures are isomorphic (as hypergraphs). To test whether two given graphs G and H have the same P4 -structure it is not necessary to first compute their P 4 -structure and then to decide whether these two hypergraphs are isomorphic. As an immediate consequence from the definition of P4 -structure the following proposition shows a simpler way to check whether two graphs have the same P 4 -structure. Proposition 1 Two graphs G and H have the same P 4 -structure if there exists a bijection ϕ : V (G) → V (H) such that four vertices {a, b, c, d} induce a P 4 in G if and only if {ϕ(a), ϕ(b), ϕ(c), ϕ(d)} induces a P 4 in H. t u The bijection ϕ used in Proposition 1 is called a P 4 -isomorphism between G and H. Figure 2 shows an example of four different graphs that all have the same P 4 structure. It is easily verified that four vertices induce a P 4 in one of these graphs if and only if the four vertices with the same labels induce a P 4 in the other three graphs. Clearly, the relation ”having the same P 4 -structure” defines an equivalence relation on the set of all graphs. The equivalence classes defined this way are called the P4 -isomorphism classes. The question arises, how difficult it is to decide whether two graphs belong to the same P4 -isomorphism class, i.e., what is the complexity of the following decision problem:

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Figure 2: Example of graphs having the same P 4 -structure. PROBLEM Input: Question:

P4 -Isomorphism Two graphs G and H Do G and H have the same P4 -structure ?

This problem is closely related to the problem GraphIsomorphism, i.e., the problem of deciding whether two given graphs are isomorphic. This problem is one of the most studied problems that has neither been shown to be in P, nor been shown to be NP-complete. See [39], [22] and [31] for more background on the graph isomorphism problem and [7], [42] for arguments why it is unlikely that GraphIsomorphism is an NP-complete problem. Reed [41] provided a simple reduction that shows that the problem P 4 -Isomorphism is polynomial time equivalent to GraphIsomorphism, i.e., a polynomial time algorithm for one of the two problems would also yield a polynomial time algorithm for the other problem. Lemma 1 (Reed 1986 [41]) P4 -Isomorphism is polynomial time equivalent to GraphIsomorphism. Sketch of proof: To transform an instance of P 4 -Isomorphism into an instance of GraphIsomorphism construct for a given graph G a bipartite graph as follows: vertices of the new graph are all vertices of G together with all induced P 4 ’s of G and two additional vertices x and y. Now connect in the new graph each vertex of G with all P4 ’s that contain this vertex and connect all P 4 ’s with x and add the edge xy. It is easily seen that two graphs are P 4 -isomorphic if and only if the bipartite graphs constructed this way are isomorphic as graphs. To transform an instance of GraphIsomorphism into an instance of P 4 -Isomorphism it is enough to add for every edge in a given graph two cycles of length at least |G|+1 that are identified in the edge of the graph. The correctness of this transformation follows from Lemma 4. t u 4


One of the central algorithmic questions concerning P 4 -structure is whether one can efficiently decide for a given 4-uniform hypergraph whether there exists a graph that has this hypergraph as its P4 -structure. PROBLEM Input: Question:

P4 -StructureRecognition A 4-uniform hypergraph H Is there a graph having H as its P 4 -structure ?

The problem P4 -StructureRecognition and its solution will be studied in more detail in Section 3 and Section 6. Its connection with the problem of recognizing perfect graphs is discussed in Section 4.

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Modules, h-Sets, Split Graphs and Unique P4 -Structure

For many P4 -structures H, we can efficiently find, not just one realization of H but all of its realizations. Indeed, as we shall see, it is the fact that we can often solve this more difficult problem which allows us to obtain a recursive algorithm to solve P4 -StructureRecognition. We cannot hope to solve this problem efficiently for all P4 -structures as some P4 -structures have exponentially many realizations. One important part of our approach is to analyze the structure of P 4 -structures with many realizations. We shall be particularly interested in characterizing graphs with unique P4 -structures. We begin our discussion in this section by studying the relationship between unique P4 -structures and modules. A module M in a graph G is a set of vertices such that no vertex outside M can distinguish between the vertices in M , i.e., every vertex outside of M is either adjacent to all or no vertices in M . A module is called trivial if it contains only one or all vertices of G. A nontrivial module is called a homogeneous set. See Figure 3 for two examples of homogeneous sets. A graph without a homogeneous set is called prime.

Figure 3: Graphs with homogeneous set (bold vertices).

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There exists a close relationship between the modules of a graph and its P 4 structure because no P4 contains a non-trivial module. Thus, a P 4 intersects a module in 0,1, or 4 vertices. More strongly we have the following: Key Fact: For any module M , every P4 is either contained in M , disjoint from M , or has exactly one vertex in M . Furthermore, if some P 4 P intersects a module M in exactly one vertex x then P − x + y is a P 4 for every vertex y in M . This is the easy direction of a classical result of Seinsche concerning graphs whose P4 -structures have no edges, the so-called P 4 -free graphs. Lemma 2 (Seinsche 1986 [43]) A graph is P4 -free if and only if every induced subgraph with at least three vertices contains a homogeneous set. t u Our Key Fact also has implications concerning the uniqueness of non-empty P 4 structures. The graph G2 in Figure 4 is obtained from the graph G 1 by replacing the subgraph induced by the vertices of M by its complement. We note that these two graphs have the same P4 -structure. In fact any two graphs one of which is obtained from the other by replacing the graph induced by a module by its complement will have the same P4 -structure. In general, these two graphs will not be isomorphic and therefore neither will have unique P 4 -structure (there are exceptions – see Figure 8). It is this fact which motivates our study of modules.

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Figure 4: Replacing a module M by its complement. The following result, which can be found in Chapter **** of this book, is the foundation on which the study of modules is built: Theorem 1 (Modular Decomposition Theorem, Gallai 1967 [21]) For any graph G = (V, E) with at least two vertices precisely one of the following conditions holds: (i) G is disconnected 6


(ii) G is disconnected (iii) There exists some Y ⊆ V, |Y | ≥ 4 and a unique partition V 1 , V2 , . . . of V such that Y induces a maximal prime subgraph in G and every V i is a module with |Vi ∩ Y | = 1. A modular decomposition of any graph can be built up by recursively applying this theorem. See Figure 5 for an example of a modular decomposition tree. Formally, the modular decomposition of G is a tree-representation of all the (possibly exponentially many) modules of G that requires only linear space. (It is the transitive reduction of the containment relation on the set of all modules which is unique up to tree isomorphisms.) 3

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Figure 5: The modular decomposition tree of a graph G. The nontrivial modules in this graph are the sets {1, 2, 3}, {1, 3}, {4, 5} and {6, 7}. The labels at the internal nodes of the decomposition tree indicate which operation of the Modular Decomposition Theorem was applied. The idea of modular decomposition goes back to Gallai [21] and is also known as substitution decomposition [35], prime tree decomposition [20] and X-join decomposition [23] and has been extended to more general combinatorial objects. The knowledge of the modular decomposition of a graph allows in several cases the efficient solution of optimization problems or the recognition problem for the underlying class of graphs (see for example [45]). It is therefore desirable to have very efficient algorithms to compute modular decompositions of graphs. In case of P4 -free graphs a linear time algorithm to compute the modular decomposition was given by Corneil, Perl and Stewart in 1985 [15]. Recently McConnell and Spinrad [34] found such an algorithm for general graphs. Their algorithm also implies a linear time recognition method for comparability graphs. Another linear time algorithm to compute the modular decomposition of graphs was presented by Cournier and 7


Habib [16]. Their algorithm is applicable to directed and undirected graphs. Efficient parallel algorithms and related algorithmic results for computing the modular decomposition can be found in [17] [18] [32]. The close relationship between modules and P 4 ’s discussed above motivates our definition of an analogous notion for P 4 -structures. A nontrivial subset S of the vertices of a 4-uniform hypergraph H is called an h-set, if every hyperedge in H intersects S in 0, 1 or 4 vertices. Furthermore, if a hyperedge h intersects S in exactly one vertex x then h − x + y is a hyperedge in H for every y in S. Obviously, if a graph contains a homogeneous set, then its P 4 -structure contains an h-set. The converse is not necessarily true as shown by an example in Figure 6. Both graphs shown in this figure have the same P4 -structure which contains an h-set, but only the graph depicted on the right contains a homogeneous set.

Figure 6: Two graphs with h-sets in the P 4 -structure. The graph that is obtained from a given graph by shrinking all maximal homogeneous sets into a single vertex is called its characteristic graph. Similarly, the hypergraph obtained by shrinking all maximal h-sets is called the characteristic hypergraph. Given a vertex x in a graph G 1 and some other graph G2 one can substitute x by G2 which means the following: Form the disjoint union of G 1 − x and G2 and connect every vertex of G2 with all neighbors of x in G1 . Clearly, in this new graph the graph G2 is a module. The following key result is an immediate consequence from the definition of h-sets and our Key Fact: Proposition 2 Let S be an h-set in a 4-uniform hypergraph H. Let x be a vertex of S. Suppose G1 is a realization of the subhypergraph induced by H − (S − x) and G2 is a realization of the subhypergraph induced by S. Then substituting G 2 for x in G1 yields a realization of G. t u The above remarks suggest that we should be particularly interested in hypergraphs which contain no h-sets. We call such hypergraphs prime. An h-set in a 4-uniform hypergraph H can easily be found in polynomial time by making use of the following algorithm [26]:

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ALGORITHM Input: Output:

Find h-Set (a, b)

A 4-uniform hypergraph H and two vertices a and b An h-set containing a and b if exists.

S := {a, b} while ∃f ∈ E(H) with |S ∩ f | ∈ {2, 3} or S ∩ f = {s} and ∃x ∈ S s. t. f − s + x 6∈ E(H) do add f to S if S 6= V (H) then output S else no h-set in H contains a and b Now, we can apply the algorithm above to determine if H has an h-set and if so apply Proposition 2 to reduce our problem to two subproblems. Repeating this process allows us to restrict our attention to prime hypergraphs. Unfortunately, there are also prime hypergraphs which do not have unique realizations. For one example, see Figure 7. The graphs in Figure 7 are split graphs, that is the vertices can be partitioned into two sets such that one of the two sets induces a clique and the other a stable set. Note that this implies that every P 4 has its endpoints in the stable set and its midpoints in the clique. It follows easily that replacing the clique by a stable set and the stable set by a clique yields a new split PSfrag replacements PSfragThe replacements graph with the same P4 -structure. next decomposition allows us to handle this and similar situations. 8 1 5 4 1

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7 6 6 Figure 7: Two prime graphs with the same P 4 -structure which are not complements of each other. The modular decomposition has been extended by Jamison and Olariu [30] to the so called homogeneous decomposition. It gives a refined decomposition of graphs by permitting the decomposition of graphs that are prime with respect to the modular decomposition. As with the modular decomposition, the homogeneous decomposition of a graph can be computed in linear time [5] and is therefore the basis for several efficient optimization algorithms, see [4] for a survey of results of this type. 9


The basis for homogeneous decomposition is the notion of p-connectedness. A graph is called p-connected, if for every partition of its vertex set into two nonempty disjoint sets, there exists a P4 in the graph that intersects both sets. A maximal pconnected subgraph is called a p-component. A p-connected graph is called separable if its vertex set can be partitioned into two nonempty sets such that each P 4 that is not completely contained in one of the two sets has its endpoints in one set and its midpoints in the other set. The foundation of the homogeneous decomposition of graphs is given by the following structure theorem: Theorem 2 (Primeval Decomposition Theorem, Jamison and Olariu 1995 [30]) For any graph G = (V, E) precisely one of the following conditions holds: (i) G is disconnected (ii) G is disconnected (iii) G is p-connected (iv) There is a unique proper separable p-component H of G with a partition (H 1 , H2 ) such that every vertex outside H is adjacent to all vertices in H 1 and misses all vertices in H2 . By recursively applying the Primeval Decomposition Theorem one gets the so called primeval decomposition tree of a graph. The homogeneous decomposition of a graph is obtained by using two more decomposition operations: one operation that allows substituting vertices by homogeneous sets and another one that gives a clique representation for split graphs. Of special interest in this context is the following result of Jamison and Olariu that shows the significance of split graphs for the problem P4 -StructureRecognition: Lemma 3 (Jamison, Olariu 1995 [30]) A p-connected graph is separable if and only if its characteristic graph induces a split graph. t u

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The Semi Strong Perfect Graph Theorem and Certifying Perfection

The Perfect Graph Theorem induces an equivalence relation on the set of all graphs by putting two graphs into the same equivalence class, if they are complements of 10


each other. The statement of the Perfect Graph Theorem then simply is that if we partition the set of all graphs into perfect and imperfect graphs, then any such equivalence class will completely belong to one side of the partition. As the P 4 is a selfcomplementary graph, this implies that the P 4 -isomorphism classes extend these equivalence classes. Chvátal [11] conjectured that the P 4 -isomorphism classes also have the property that their members are either all perfect or all are imperfect. This conjecture was called the Semi Strong Perfect Graph Conjecture. Chvátal was led to this conjecture by observing that odd cycles of length at least five and their complements have unique P4 -structure, i.e., the P4 -isomorphism class of these graphs contain only the graph and its complement. Hayward then observed [24] that the same also holds for long enough even cycles, so we have: Lemma 4 (Chv´ atal 1984 [11], Hayward 1983 [24]) Odd cycles of length at least five and even cycles of length at least 8 have unique P4 -structure. t u Note that the C6 does not have unique P4 -structure as all graphs in Figure 2 have the P4 -structure of a C6 . The Strong Perfect Graph Conjecture together with Lemma 4 implies the truth of the Semi Strong Perfect Graph Conjecture. On the other hand the Perfect Graph Theorem is implied by the Semi Strong Perfect Graph Conjecture, which explains why it was given this name. The Semi Strong Perfect Graph Conjecture first has been proved for the special cases of bipartite graphs [14] and triangulated graphs [24] until in 1987 Reed gave a proof for the conjecture in general which was from then on called the Semi Strong Perfect Graph Theorem. Theorem 3 (The Semi Strong Perfect Graph Theorem, Reed 1987 [40]) A graph is perfect if and only if it has the P 4 -structure of a perfect graph. t u This result shows that for solving the recognition problem for perfect graphs, i.e. the problem PROBLEM PerfectGraphRecognition Input: A graph G Question: Is G perfect ? it suffices to be able to identify P4 -structures that belong to perfect graphs. This problem is called PerfectP4 -StructureRecognition: 11


PROBLEM Input: Question:

PerfectP4 -StructureRecognition A 4-uniform hypergraph H Is there a perfect graph having H as its P 4 -structure ?

Chvátal[13] asked whether there exists a polynomial time algorithm for the problem PerfectP4 -StructureRecognition. Clearly, such an algorithm would yield a polynomial time algorithm for the problem of recognizing perfect graphs. Therefore, Chvátal also asked for a simpler problem, namely the recognition of P 4 -structures for special classes of (perfect) graphs. Such a result could lead to an NP characterization of perfect graphs as follows. To certify the perfection of a given graph G it is enough to present a graph H for which its perfectness is easily verified and that has the same P4 -structure as G. The question of course arises whether such a graph H always exists. There are many classes of graphs for which we know the perfectness as for example bipartite graphs or triangulated graphs. Therefore a natural question is whether there exists a simple class C of perfect graphs that intersects every P4 -isomorphism class that contains only perfect graphs. Question 1 Is there some simple class C of perfect graphs such that for each perfect graph G there is a graph in C that has the same P 4 -structure as G ? Clearly, a positive answer to Question 1 would imply that the problem PerfectGraphRecognition belongs to NP. We may assume that a class C that could be an answer to Question 1 is closed under taking graph complements, as these can be computed in polynomial time. Therefore, a necessary condition for a class C is that it must contain all graphs that have unique P 4 -structure. This leads us to the next question: Question 2 Which (perfect) graphs have unique P 4 -structure ? We have already touched on this question in Section 3, we will discuss it further in the next section. This discussion implies that we get a negative answer to Question 1.

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The Structure of the P4 -Isomorphism Classes

To answer Question 2 we need to know which (perfect) graphs have unique P 4 structure. First we will mainly be interested in asymptotic results which will tell us how many (perfect) graphs have unique P 4 -structure. For a class C of graphs and a graph property Π we say that almost all graphs in C have property Π if the ratio |Cn (Π)|/|Cn | tends to 1, as n goes to infinity. Here C n denotes all graphs in C on n vertices and Cn (Π) denotes all graphs in Cn that have property Π. It is easily seen that the P4 -isomorphism classes can be arbitrarily large, as for 12


example all P4 -free graphs belong to the same P4 -isomorphism class. Thus, the following result may be surprising at the first sight, but a close look at Reed’s proof of the Semi Strong Perfect Graph Theorem [40] reveals that a weaker form of the following result is essentially the idea of the proof. Theorem 4 (Hougardy 1996 [28]) Almost all graphs have unique P4 -structure.

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This result shows that the P4 -isomorphism classes almost always contain just two elements, namely the graph and its complement. But, as almost no graph is perfect, it could still be the case that on the class of perfect graphs these equivalence classes have almost always a much larger size. The following result shows that this is not the case. Theorem 5 (Hougardy 1996 [28]) Almost all perfect graphs have unique P 4 -structure.

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The proof of this result is based on a result of Prömel and Steger [38] who proved that almost all Berge graphs are perfect, i.e., they showed that the Strong Perfect Graph Conjecture is almost always true. As noted in Section 3 there exists a close relation between the P 4 -structure of graphs and homogeneous sets. As we saw in Section 3, most graphs containing a homogeneous set do not have unique P4 -structure. Some exceptions are shown in Figure 8 We need a slightly more restricted notion of unique P4 -structure to get the desired connection between homogeneous sets and unique P4 -structure. A graph is said to have strongly unique P4 -structure if it has unique P4 -structure and every P4 -automorphism is also a graph automorphism.

Figure 8: Graphs with homogeneous set (bold vertices) and unique P 4 -structure.

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Clearly, if a graph has strongly unique P 4 -structure it also has unique P4 -structure. The C5 is an example of a graph that has unique but not strongly unique P 4 -structure: each of the 120 permutations of the five vertices is a P 4 isomorphism, but only 10 of them are graph isomorphisms. If a graph G has a homogeneous set S and we take the complement on the graph induced by S, this will give us a P4 -isomorphism but not a graph isomorphism or an isomorphism to the complement of G. Thus, if a graph has a homogeneous set it cannot have strongly unique P4 -structure. The other direction does not hold in general, as for example the P5 or C6 provide examples of graphs without homogeneous set which do not have strongly unique P 4 -structure. But it turns out, that graphs without homogeneous set that contain a certain fixed induced subgraph all have strongly unique P4 -structure. Theorem 6 (Hougardy 1996 [28]) If a graph G contains an induced cycle of length at least seven then G has strongly unique P4 -structure if and only if G does not contain a homogeneous set. t u Theorem 6 was generalized by Hayward, Hougardy and Reed, as follows. A graph is CSC if it can be partitioned into two cliques with no edges between them and a stable set. It is SCS if its complement is CSC. Theorem 7 (Hayward, Hougardy, Reed 1996 [26]) If a prime graph G contains a (prime) subgraph F which has a strongly unique P 4 structure then provided F is neither SCS nor CSC, G also has a strongly unique P4 -structure. The example of Figure 9 shows that the condition that F is not SCS or CSC is necessary. All the graphs of Figure 10 are neither SCS nor CSC and all have strongly unique P4 -structure. Thus Theorem 6 remains true if we replace cycle of length seven by any graph in Figure 10. Theorem 7 has negative consequences for the applicability of P 4 -structure to certify perfection: for all perfect graphs that contain this subgraph, P 4 -structure is no better than homogeneous sets in certifying perfection. Theorem 6 and 7 show that the class C that was asked for in Question 1 cannot exist. On the other hand as we see in the next section, Theorem 7 is very helpful in designing an algorithm to solve P4 -StructureRecognition.

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Figure 9: A prime graph without strongly unique P 4 -structure even though the subgraph induced by the bold vertices has strongly unique P 4 -structure.

Figure 10: Graphs with strongly unique P 4 -structure that are neither SCS nor CSC.

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Recognizing P4 -Structure

The preceding sections have shown how the problem P 4 -StructureRecognition relates to the problem PerfectGraphRecognition. While the latter problem is still open, the problem P4 -StructureRecognition allows a polynomial time algorithm for its solution as was shown by Hayward, Hougardy and Reed: Theorem 8 (Hayward, Hougardy, Reed 1996 [26]) The problem P4 -StructureRecognition can be solved in polynomial time.

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Theorem 7 is the key to the algorithm of Hayward, Hougardy and Reed. They also need the following results: Lemma 5 If F is a prime hypergraph of a prime P 4 -structure H then there is a prime subhypergraph of H containing F and at most two other vertices. t u This allows them to solve P4 -StructureRecognition, given a strongly unique realization of a subhypergraph F of H which is neither SCS or CSC as follows: They repeatedly find a larger prime hypergraph by applying Lemma 5 and find the corresponding unique realization (it turns out to be fairly straightforward to find all the 2 vertex extensions of a given realization). 15


The rest of the algorithm involves dealing with hypergraphs for which we cannot find such a subhypergraph F. One crucial point is to have a subroutine that checks whether a 4-uniform hypergraph H has a realization as a split graph. Hayward, Hougardy and Reed [26] present a polynomial time algorithm for this problem restricted to h-set free hypergraphs and show how it can be extended to hypergraphs containing h-sets. Their algorithm starts with the realization of one single hyperedge partitioned into four sets W, X, Y, Z and successively adds in each step one or two vertices to the sets W, X, Y, Z such that the following two conditions hold • In any realization of H as a split graph the vertices from the same set must belong to the same side of the split graph • There exists a P4 with a vertex in each of W, X, Y, Z. When posing the problem P4 -StructureRecognition Chvátal [13] also asked whether this problem can be solved for some interesting subclasses of (perfect) graphs. PROBLEM Input: Question:

P4 -StructureRecognitionForClassC A 4-uniform hypergraph H Is there a graph in C having H as its P 4 -structure ?

The first class C for which a polynomial time algorithm for the problem P 4 StructureRecognitionForClassC was presented was the class of trees. Ding [19] gave such an algorithm whose running time was later improved by Brandstädt, Le and Olariu [9]. Meanwhile the problem P 4 -StructureRecognitionForClassC has been proved for many other classes of (perfect) graphs. For example Babel, Brandstädt and Le solved the problem for bipartite graphs [3], Brandstädt and Le for block graphs [8], Sorg for linegraphs of bipartite graphs [44], Babel for linegraphs [1] and for claw-free graphs [2], and Hayward, Hougardy and Reed for split graphs [26].

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The P4 -Structure of Minimally Imperfect Graphs

A graph is called unbreakable if neither the graph nor its complement contains a star-cutset. Chvátal’s Star-Cutset Lemma states that minimally imperfect graphs are unbreakable. To prove the Strong Perfect Graph Conjecture it is therefore enough to have it proved for the class of unbreakable graphs. This explains why studying the P4 -structure of unbreakable graphs is of special interest with respect to perfect 16


graphs. There exist several results that show that the P 4 -structure of an unbreakable graph is in some sense densely connected. Two P4 ’s h and h0 in a graph G are 3-chained if there exists a sequence h = h1 , h2 , . . . , hk = h0 of P4 ’s in G such that any two consecutive ones have exactly three vertices in common. Lemma 6 (Chv´ atal 1987 [12]) In an unbreakable graph every two P4 ’s are 3-chained.

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An alignment in a graph is a sequence Q 1 , Q2 , . . . , Qk of sets of vertices such that each Qi induces a P4 , and each Qi with i ≥ 2 has precisely one vertex outside Q1 ∪ Q2 ∪ . . . ∪ Qi−1 . The alignment is called full if each vertex of the graph belongs to at least one Qi . Lemma 7 (Chv´ atal, Ho` ang 1985 [14]) In an unbreakable graph every alignment extends into a full alignment.

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Both these results have been extended by Olariu [36]. He partitions the edges of a graph G into coercion classes such that two edges xy and ab belong to the same coercion class if abxy is a P4 in G. Theorem 9 (Olariu 1991 [36]) An unbreakable graph contains at most two coercion classes and in an unbreakable graph with exactly two coercion classes the neighborhood of any vertex is the disjoint union of two cliques. t u Any counterexample to the Strong Perfect Graph Conjecture must be C 5 -free and unbreakable which makes this class of graphs of special interest. For the class of C5 -free unbreakable graphs we can give a complete description of the structure of the P4 -isomorphism classes. Theorem 10 (Hougardy 1997 [29]) C5 -free unbreakable graphs different from C 6 and its complement have unique P4 structure. t u This result shows that for the class of C 5 -free unbreakable graphs the Semi Strong Perfect Graph Theorem and the Perfect Graph Theorem make exactly the same statement. 17


Theorem 10 cannot be extended to the class of all unbreakable graphs as there exist unbreakable graphs different from C 6 and its complement that have the same P4 -structure. Figure 11 shows two such graphs. PSfrag replacements PSfrag replacements 2 1

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Figure 11: Two unbreakable graphs with the same P 4 -structure From Theorem 10 one can easily derive the Semi Strong Perfect Graph Theorem. However, it should be noted, that the proof of Theorem 10 is longer than the proof of the Semi Strong Perfect Graph Theorem, thus the following proof is not really a better proof to the Semi Strong Perfect Graph Theorem but sheds some more light on it. Short proof for the SSPGT Assume G and H are two graphs having the same P 4 -structure such that G is perfect and H is minimally imperfect. Chvátal’s Star-Cutset Lemma implies that H is C 5 free and unbreakable. Now Theorem 10 shows that H has unique P 4 -structure, i.e. G or its complement must be isomorphic to H, which is a contradiction. t u

8

The Partner Structure and Other Generalizations

We have seen in Section 5, that the P4 -isomorphism classes almost always are of size at most two. Thus, a natural question arises whether the notion of P 4 -structure can be extended in such a way that the equivalence classes defined by this extension become larger and still a statement similar to the Semi Strong Perfect Graph Theorem holds. One such approach uses the notion of partner structure. Two vertices a and b of a graph G are called partners if there are vertices x, y, z in G − {a, b} such that {a, x, y, z} and {b, x, y, z} each induce a P 4 in G. The partner 18


graph of a graph G is the graph whose vertices are the vertices of G, and whose edges are the pairs of vertices that are partners in G. The notion of partner graphs was introduced by Chvátal [12] who proved the following beautiful decomposition result: Lemma 8 (Chv´ atal [12] 1987) If the vertices of a graph G can be colored by two colors such that any two partners obtain the same color, then G is perfect if and only if the two graphs induced by the two color-classes are perfect. To test whether a given graph admits a two-coloring as requested in the lemma, one simply has to check whether its partner-graph is connected. Therefore a decomposition according to Chvátal’s result can be found in polynomial time. However, not all perfect graphs admit such a decomposition. Motivated by Chvátal’s result Hoàng [27] defined two graphs G and G 0 to have the same partner-structure if there exists a mapping f : V → V 0 such that vertices x, y are partners with respect to a set {a, b, c} in G if and only if f (x), f (y) are partners with respect to f ({a, b, c}) in G 0 . Thus, the partner-structure of a graph is obtained by removing all hyperedges from its P 4 -structure that do not intersect some other hyperedge in exactly three vertices. Obviously, if two graphs have the same P4 -structure then they also have the same partner-structure. The other direction does not hold as can be seen for example by considering the P 4 and the K4 . Thus, the following result of Hoàng is a generalization of the Semi Strong Perfect Graph Theorem. Its relation to the Semi Strong Perfect Graph Theorem is studied in [28]. Theorem 11 (Ho` ang 1990 [27]) 0 Let G and G be two graphs with the same partner-structure. Then G is perfect if and only if G0 is. t u

9

P3 -Structure

The notion of P4 -structure can be extended to F -structure for arbitrary fixed graphs F . Thus, it is a natural question to ask what is so special about the P 4 that makes it closely related to perfect graphs allowing results like the Semi Strong Perfect Graph Theorem ? As we will see in this section, there exists only one other graph, namely the P3 , that leads to results similar to the Semi Strong Perfect Graph Theorem. For a fixed graph F , all graphs not containing F as an induced subgraph have the same F -structure. Thus any graph that is a candidate to replace the P 4 in the 19


Semi Strong Perfect Graph Theorem must be an induced subgraph of all sufficiently long (i.e. of size at least |F |) odd holes and odd antiholes. This implies that the stability number and the clique number of F must have the value at most two. Thus the only reasonable candidate for a statement similar to the Semi Strong Perfect Graph Theorem isPSfrag the P3replacements or its complement. However, it is easily seen that 1 on its P 3 -structure. Figure 12 shows the perfection of a graph does not just depend a well known counterexample. 2 3 1 4 5 PSfrag replacements 2

5

5

4 2

4

1

3

3

Figure 12: Two graphs with the same P 3 -structure. This simple counterexample leads astray as it seems to suggest that for larger graphs there exist much more counterexamples. However, one can show that this simple example is essentially the only exceptional case. More precisely we have Theorem 12 (Hougardy 1996 [28]) A C5 -free graph is perfect if and only if it has the P 3 -structure of a perfect graph.

t u

In contrast to the P4 the P3 is not a self-complementary graph. Therefore Theorem 12 does not immediately imply the Perfect Graph Theorem. But using the Perfect Graph Theorem one easily gets the following stronger result. Theorem 13 (Hougardy 1996 [28]) A C5 -free graph G is perfect if and only if G or G has the P 3 -structure of a perfect graph. t u Of course Theorem 13 implies the Perfect Graph Theorem, and the following result shows that the Strong Perfect Graph Conjecture implies Theorem 13. Therefore Theorem 12 can be seen as ”another” Semi Strong Perfect Graph Theorem. Theorem 14 (Hougardy 1996 [28]) Cycles of length at least six and complements of odd cycles of length at least seven have unique P3 -structure. t u 20


Even so Theorem 12 lacks the beauty of the Semi Strong Perfect Graph Theorem, it has in principle the same consequences as the Semi Strong Perfect Graph Theorem. A certificate to prove perfection for a given graph now would just consist of two parts. The first part is a perfect graph that has the same PPSfrag as the given graph. 3 -structure replacements The second part is the proof thatPSfrag the given graph does not contain a C 5 as an induced replacements 1 subgraph. This is obviously easily done in polynomial time. 1 2 Since the P3 is a subgraph of the P4 , one might2 think that Theorem 313 is already implied by the Semi Strong Perfect Graph Theorem. 3 That this is not the4 case can be shown by the graphs in Figure 13a) and Figure 13b).4 This is a non-trivial5 example of 5 6 the other PSfrag replacements two graphs that have the same P3 -structure but different P4 -structure. On hand there (of course) exist examples of graphs that6 have the same P 4 -structure but 2 PSfrag replacementsdifferent P - and P -structure. One example is shown in Figure 13c) and Figure 13d). 2 2 3 3 3 3 3 6

1 3 2

4

5

6 1

a)

5

2

4

4 3 5

3 b)

2 c)

1

3 4

3 5

2

1

d)

Figure 13: Two pairs of graphs showing that P 3 -structure and P4 -structure are independent of each other. Not very surprisingly, the asymptotic results that were stated in Section 5 for P4 -structure also hold for P3 -structure. Theorem 15 (Hougardy 1996 [28]) Almost all (perfect) graphs have unique P 3 -structure.

t u

The Problem P4 -StructureRecognition can be naturally extended from P 4 to any other graph F . PROBLEM F -StructureRecognition Input: A |F |-uniform hypergraph H Question: Is there a graph having H as its F -structure ? Currently, the P3 is the only non-trivial graph different from the P 4 for which the complexity of this problem is known: Theorem 16 (Hayward 1996 [25]) 21


There exists a polynomial time algorithm to recognize the P 3 -structures of graphs.

t u

Acknowledgement I am grateful to Bruce Reed for useful comments on this paper.

References [1] L. Babel, On the P4 -Structure of Line Graphs, Technical Report, TUM-M9807, Zentrum Mathematik, Technische Universität M¨ unchen, 1998 [2] L. Babel, Recognizing the P4 -structure of claw-free graphs, technical report [3] L. Babel, A. Brandstädt, V.B. Le, Recognizing the P 4 -structure of bipartite graphs, Discrete Applied Mathematics 93 (1999), 157–168 [4] L. Babel, S. Olariu, On the p-connectedness of graphs - a survey, Discrete Applied Mathematics 95 (1999), 11-33 [5] S. Baumann, A linear algorithm for the homogeneous decomposition of graphs, Report 96-15, Technical University Munich 1996 [6] R.G.Bland, H.-C. Huang, L.E.Trotter Jr., Graphical properties related to minimal imperfection, Annals of Discrete Mathematics 21 (1984), 181–192 [7] R.B.Boppana, J.Hastad, S.Zachos, Does co-NP have short interactive proofs ?, Inform. Process. Lett. 25 (1987), 127–133 [8] A. Brandstädt, V.B. Le, Recognizing the P 4 -structure of block graphs, Manuskript Univ. Rostock, 1997, to appear in Discrete Applied Mathematics [9] A. Brandstädt, V.B. Le, S. Olariu, Efficiently recognizing the P 4 -structure of trees and of bipartite graphs without short cycles, to appear in Graphs and Combinatorics [10] K.Cameron, Polyhedral and algorithmic ramifications of antichains, PhD Thesis, University of Waterloo (1982) [11] V. Chvátal, A semi-strong perfect graph conjecture, Annals of Discrete Mathematics 21 (1984), 279–280

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[12] V. Chvátal, On the P4 -structure of perfect graphs III. Partner decompositions, Journal of Combinatorial Theory, Series B 43 (1987), 349–353 [13] V. Chvátal, Problems concerning perfect graphs, Manuscript from the Princeton Conference on Perfect Graphs, Princeton 1993 [14] V. Chvátal, C.T.Hoàng, On the P 4 -structure of perfect graphs I. Even decompositions, Journal of Combinatorial Theory, Series B 39 (1985), 209–219 [15] D.G. Corneil, Y. Perl, L.K.Stewart, A linear recognition algorithm for cographs, SIAM J. Comput. 3 (1985), 926–934 [16] A.Cournier, M.Habib, A new linear algorithm for modular decomposition, in: S. Tison (Ed.) CAAP’94: 19th Internat. Coll. Edinburgh, UK, Lecture Notes in Computer Science, Springer, Berlin 1994, 68–82 [17] E. Dahlhaus, Efficient parallel modular decomposition, in: 21st International Workshop on Graph-Theoretic Concepts in Computer Science 1995, 290–302 [18] E. Dahlhaus, J.Gustedt, R.McConnell, Efficient and practical modular decomposition, Eighth Annual ACM-SIAM Symposium on Discrete Algorithms 1997, 26–35 [19] G.Ding, Recognizing the P4 -structure of a tree, Graphs Combin. 10 (1994), 323–328 [20] A. Ehrenfeucht, G. Rozenberg, Theory of 2-structures, part 1: clans, basic subclasses, and morphisms, Theoret. Comput. Sci 70 (1990), 277–303 [21] T. Gallai, Transitiv orientierbare Graphen, Acta Mathematica Academiae Scientarum Hungaricae 18 (1967), 25–66 [22] G. Gati, Further annotated bibliography on the isomorphism disease, Journal of Graph Theory 3 (1979), 95–109 [23] M. Habib, M.C.Maurer, On the X-join decomposition for undirected graphs, Discrete Appl. Math. 1 (1979) 201–207 [24] R.B. Hayward, On the P4 -structure of perfect graphs III. Weakly triangulated graphs, Technical Report McGill University S.O.C.S.-83.22, October 1983 [25] R.B. Hayward, Recognizing P3 -structure: A switching approach, Journal of Combinatorial Theory, Series B 66 247–262 (1996)

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[26] R.B. Hayward, S. Hougardy, B. A. Reed, Recognizing P 4 -structure, Technical Report 1996 [27] C.T. Hoàng, On the sibling structure of perfect graphs, J. Combin. Theory Ser. B 49 (1990), 282–286 [28] S. Hougardy, On the P4 -structure of perfect graphs, PhD thesis, Shaker Verlag, Aachen 1996 [29] S. Hougardy, Perfect graphs with unique P 4 -structure, Discrete Mathematics 165/166 (1997), 421–430 [30] B. Jamison, S. Olariu, P-components and the homogeneous decomposition of graphs, SIAM J. Disc. Math. 8 (1995), 448–463 [31] D.S. Johnson, A catalog of complexity classes, in: Handbook of Theoretical Computer Science, Volume A (J. van Leeuwen eds), MIT Press/Elsevier 1990 [32] R. Lin, S. Olariu, A fast parallel algorithm to recognize P 4 -sparse graphs, Discrete Appl. Math. 81 (1998), 191–215 [33] L. Lovász, A characterization of perfect graphs, Journal of Combinatorial Theory, Series B 13 (1972), 95–98 [34] R.M.McConnell, J.P.Spinrad, Modular decomposition and transitive orientation, Discrete Mathematics 201 (1999), 189–241 [35] R.H. Möhring, Algorithmic aspects of comparability graphs and interval graphs, in: I. Rival (Ed.), Graphs and Orders, D.Reidel, Boston 1985, 41–101 [36] S. Olariu, On the structure of unbreakable graphs, Journal of Graph Theory 15 (1991), 349–373 [37] M.W.Padberg, Perfect zero–one matrices, Math. Programming 6 (1974), 180– 196 [38] H.J.Prömel, A.Steger, Almost all Berge graphs are perfect, Combinatorics, Probability and Computing (1992) 1, 53–79 [39] R.C. Read, D.G.Corneil, The graph isomorphism disease, Journal of Graph Theory 1 (1977), 339–363 [40] B.A.Reed, A semi-strong perfect graph theorem, Journal of Combinatorial Theory, Series B 43 (1987), 223–240

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[41] B.A.Reed, A semi-strong perfect graph theorem, PhD thesis, McGill University, 1986 [42] U. Schöning, Graph Isomorphism is in the low hierarchy, in: Proc. 4th Symp. on Theoretical Aspects of Computer Science, LNCS 247, Springer1986, 114–124 [43] D.Seinsche, On a property of the class of n–colorable graphs, Journal of Combinatorial Theory, Series B 16 (1974), 191–193 [44] S. Sorg, Die P4 Struktur von Kantengraphen bipartiter Graphen, Diplomarbeit, Universität zu Köln, 1997 [45] J.P.Spinrad, P4 -trees and substitution decomposition, Discrete Appl. Math. 39 (1992), 263–291

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